Question

random sample of 16 observations from a normal population. The sample standard deviation = 5 and the sample mean = 100. In testing the null hypothesis that µ = 110 against the alternative hypothesis that µ < 110. Which test statistic should be used?

Answer 1

The test statistic (Z) in this case is -8.

To test the hypothesis that the population mean (μ) is less than 110 using a sample with known values, you can use a one-sample z-test because the sample size is large (n = 16) and the population standard deviation is unknown.

The formula for the z-test statistic when the population standard deviation is unknown is:

`Z= (X- \mu_0)/((s)/(√(n)))`

​Where:

X is the sample mean (given as 100).

μ_0 is the hypothesized population mean under the null hypothesis (given as 110).

s is the sample standard deviation (given as 5).

n is the sample size (given as 16).

Substituting the values:

`Z= (110 -100)/((5)/(√(n) ) )`

`Z= (-10)/((5)/(4) ) = (-10)/(1.25) = -8`

The test statistic (Z) in this case is -8. This value would then be used to determine the probability (p-value) of observing such an extreme value under the assumption that the null hypothesis (μ=110) is true.